An online database of classes of algebraic structures
Chapman University, CA
ASL 2003 Annual Meeting, June 1-4, 2003
A currently ongoing research project
is the construction of a web
database of classes of mathematical structures
The aim of this project is to make basic information
mathematical structures available in a uniform and extendible way
with direct connections to computational tools and decision
for these structures.
- Two examples that influenced this project
- A look at the current version of the database
classes of mathematical structures have
been investigated in the last century.
Even if we restrict to
classes related to algebraic logic
there are still over 100 classes
that have been defined and analyzed
in some detail.
, model theory
and category theory
have been developed to express general properties and analyze relations
between classes of mathematical structures.
But the information known about individual classes
of structures is
still much greater
than what is covered by general results
in universal algebra, logic
or category theory
Classes of mathematical structures
can be defined in several
different ways, e.g. by
1) specifying a list of properties
that they must satisfy, often
expressed as formulas of (first-order) logic, or
2) specifying a generating class of algebras
closure operator (e.g. HSP)
It can be nontrivial
to determine whether a general result
applies to a specific class
Especially for researchers who are working in other areas
, it is
often difficult to assess whether a particular class of structures
(or a closely related one) has already been examined
in detail in a
The aim of this online database of mathematical structures
address the problem by providing broad coverage
of the many
classes of structures that have been investigated in the literature.
The aim is not to concentrate
only on the major classes, but to
include basic information also on specialized less well-known
How can such a database be constructed?
Two examples that influenced this project
Exhibit A: http://www.wikipedia.org
- three years in the making (so far)
- over 100000 up-to-date entries, including reliable information about general mathematical research topics
- designed collaboratively on a Wiki server platform
- no copyright; material can be freely copied
- very low development cost; surprisingly high quality
Neil Sloane's Encylopedia of integer sequences
- sequences collected by N. Sloane over the past few decades
- over 60000 sequences (mostly initial segments) related to mathematical structures
- some computational tools and search tools for finding related sequences
- used by many researchers on a daily basis
- new entries submitted by email (moderated)
There are several other databases, e.g. Wolfram's Mathworld,
These resources are designed for humans to read. They are extensive, high
quality, up-to-date, freely available, and getting better all the time.
The current mathematical structures database project
somewhere between Sloane's integer sequences (which give some invariants for
classes of mathematical structures) and Mathworld or Wikipedia.
An emphasis is on making data easily available to software
computes further mathematical information about the classes.
Several researchers have used computers for calculations on finite algebras,
e.g. C. Bergman, J. Berman, S. Burris, S. Comer, R. Freese, E. Kiss, E.
Lukacs, R. Maddux, M. Maroti, J.B. Nation, S. Tschantz, M. Valeriote, ...
This database aims to make the results of such calculations easily
Initially, each class of structures has a webpage in the database containing
(some of) the following items:
- Definition (usually several equivalent ones)
- Description of the morphisms
- Standard examples
- Basic results (or references to the literature)
- List of relevant properties (with values and references)
- Information on free objects, injective, projectives, etc.
- List of some (finite) members of the class, possibly with some graphical representations
- List of subclasses (or subcategories)
- List of superclasses (or supercategories)
A format for specifying axioms (to be extended
to theorems and proofs)
Currently based on TeX+XML, to be converted to MathML
An XML definition for finite
structures (to be extended to automatic structures and finitely presented
Linking algebraic and logical viewpoints.
But first: What is XML
A (tree) structured
, human and
, platform independent
to understand; e.g. many students know it
because HTML is
a relaxed form of XML.
Well suited for mathematical expressions
(e.g. MathML is a developing XML standard)
E.g. here is an XML description of a 3-element poset:
<structure name="Poset_3_4" size="3">
<element id="0" x="1" y="0"/>
<element id="1" x="0" y="1"/>
<element id="2" x="2" y="1"/>
<relation name="uppercovers" arity="2" type="list">
<set id="0"> 1 2 </set>
<set id="1"> </set>
<set id="2"> </set>
Design guidelines for the database:
- Natural syntax (modelled on mathematical practice)
- Collaborative development style
- Accessible algorithms (open architecture)
- Interactive diagrams
- Interactive diagrams
- Platform independent
- Useable as a teaching/learning tool
For this presentation we will briefly consider the following algebraic
- Kleene algebras
- action algebras
- residuated lattices
- Boolean semigroups
So how does it look at the moment?
Go to math.chapman.edu/structures
(or search for Mathematical Structures Homepage
Standardize the XML format for finite structures, write a DTD.
Design an XML format for declarative mathematical proofs based on the
defacto syntax of mathematics (e.g. set theory + TeX).
tools for proof checking and proof generation.
A format for finitely presented algebras, partial algebras, multi-sorted
Better webdesign, MathML support, TeX output.
More graphical representations of algebras (e.g. Cayley graphs).
Add more classes, references and results to the database
(without duplicating efforts of other webdatabases).
The database project is still developing
, but it is already useful.
There are a number of new ideas implemented in it that are not found in
other online resources:
- platform independent format for mathematical structures
- direct support for simple algorithms
- interactive graphical support
- collaborative development style.
It represents a form of instant peer-reviewed anonymous publishing
Every class of mathematical structures that has been
investigated (has some publications) deserves a presence on the web.
As the database becomes more complete, the interesting gaps
will be more obvious.
This project will not be finished at some stage. If it succeeds, it will
continue to evolve and remain up to date.
. --- Some things are surprisingly
Storage space is not a problems. It's like the wild west all over again.
If you are happily farming on a little plot in New Jersey, watch out, there
are other subjects that a claiming the huge midwestern storage plains, and
they gain intrinsic status by being able to manage big databases.
I'm not advocating that we produce huge amounts of worthless
data --- mathematics has many legitimate reasons for generating lots of
useful data. E.g. lists of finite structures can be a good source of
conjectures and counterexamples.
I hope you are upset
that your favorite category or class of
is not mentioned or has very little information in the database
This makes it more likely
that someone will add it.
It only takes half an hour
to make a significant contribution
So encourage your students
to do it.
How often do we find annoying little errors
in papers or on webpages?
Here you can fix them immediately
The database will remain freely accessible
. Anyone can get a
(or search for Mathematical Structures Homepage
, or my homepage on the web)